Asymptotic experssion of the solution of general boundary value problem for higher order elliptic equation with perturbation both in boundary and in operator

In this paper basing on (1) and (2), we study the singular perturbation of general boundary value problem for higher order elliptic equation with perturbation both in the boundary and in the operator, so as to establish the asymptotic expression involving two parameters. Thus, the iterative process of finding the asymptotic solution is derived and the estimation of the remainder term is given out, we extend and improve the previously published papers.     

Free vibration of a class of Hill's equation having a small parameter

In this paper, we consider the initial value problem of a class of Hill’s equation having a small parameter. Using the solvable condition of boundary value problem and the stretched parameter method in the perturbation techniques, we present the method which can be applied to obtain asymptotic periodic solution of the initial value problem. As an example, we consider Mathieu equation and present its computational result.     

弹性板边界元中计算边界高阶导数的渐近收敛积分方程

本文针对板弯曲边界元方法中计算边界曲率等高阶导数项时边界积分方程中出现的高阶奇异积分项,通过对未知挠曲函数作渐近展开并加以适当摄动,获得了渐近收敛的边界积分方程。采用这一方法计算板边界上的曲率分布,获得了满意的数值结果。     

A singular perturbation problem for periodic boundary partial differential equation

In this paper,we consider a singular perturbation elliptic-parabolic partial differentialequation for periodic boundary value problem,and construct a difference scheme.Using themethod of decomposing the singular term from its solution and combining an asymptoticexpansion of the equation,we prove that the scheme constructed by this paper convergesuniformly to the solution of its original problem with O(τ h~2).     

Asymptotic Stability of Rarefaction Wave for the Navier–Stokes Equations for a Compressible Fluid in the Half Space

This paper is concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier–Stokes equations in a compressible fluid in the Eulerian coordinate in the half space. This is the second one of our series of papers on this subject. In this paper, firstly we classify completely the time-asymptotic states, according to some parameters, that is the spatial-asymptotic states and boundary conditions, for this initial boundary value problem, and some pictures for the classification of time-asymptotic states are drawn in the state space. In order to prove the stability of the rarefaction wave, we use the solution to Burgers’ equation to construct a suitably smooth approximation of the rarefaction wave and establish some time-decay estimates in L ^p -norm for the smoothed rarefaction wave. We then employ the L ²-energy method to prove that the rarefaction wave is non-linearly stable under a small perturbation, as time goes to infinity. P. Zhu was supported by JSPS postdoctoral fellowship under P99217.     

A Perturbation Argument for a Monge–Ampère Type Equation Arising in Optimal Transportation

We prove some interior regularity results for potential functions of optimal transportation problems with power costs. The main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost, but it does not satisfy the well known condition (A.3) guaranteeing regularity. The proof consists in a perturbation argument from the standard Monge–Ampère equation in order to obtain, first, interior C^1,1 estimates for the potential and, second, interior Hölder estimates for second derivatives. In particular, we take a close look at the geometry of optimal transportation when the cost function is close to quadratic in order to understand how the equation degenerates near the boundary.     

Exponential dichotomies of nonlinear discrete systems and its application to numerical analysis and computation

In this pepar we consider the upwind difference scheme of a kind of boundary value problems for nonlinear, second order, ordinary differential equations. Singular perturbation method is applied to construct the asymptotic approximation of the solution to the upwind difference equation. Using the theory of exponential dichotomies we show that the solution of an order-reduced equation is a good approximation of the solution to the upwind difference equation except near boundaries. We construct correctors which yield asymptotic approximations by adding them to the solution of the order-reduced equation. Finally, some numerical examples are illustrated.     

Asymptotic Solution for a Kind of Boundary Layer Problem

This paper studies the partial differential equation with a small coefficient in the highest-order item. This kind of equation is also named as boundary layer problem. The Burgers equation and modified Burgers equation are analyzed in this approach. First, these equations are transferred into the strong nonlinear ones, and then the corresponding strong nonlinear equations are solved based on the perturbation method. The results from the asymptotic method are comparable with those obtained from numerical computation. An erratum to this article is available at .     

Thin elastic coatings and surface enthalpy

We obtain explicit formulas for the first few terms in the asymptotics of the solution of the problem on the deformation of a body with a thin elastic coating and propose several methods, based on these formulas, for modeling the problem with the use of the Wentzel boundary conditions or a regular perturbation of the boundary. We present estimates of the asymptotic remainders, including those in formulas obtained by modeling.     

Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation

In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed.     

Steady flows of non-Newtonian fluids past a porous plate with suction or injection

The problem of the steady flow of three classes of non-linear fluids of the differential type past a porous plate with uniform suction or injection is studied. The flow which is studied is the counterpart of the classical ‘asymptotic suction’ problem, within the context of the non-Newtonian fluid models. The non-linear differential equations resulting from the balance of momentum and mass, coupled with suitable boundary conditions, are solved numerically either by a finite difference method or by a collocation method with a B-spline function basis. The manner in which the various material parameters affect the structure of the boundary layer is delineated. The issue of paucity of boundary conditions for general non-linear fluids of the differential type, and a method for augmenting the boundary conditions for a certain class of flow problems, is illustrated. A comparison is made of the numerical solutions with the solutions from a regular perturbation approach, as well as a singular perturbation.     

Boundary Perturbations Due to the Presence of Small Linear Cracks in an Elastic Body

In this paper, Neumann cracks in elastic bodies are considered. We establish a rigorous asymptotic expansion for the boundary perturbations of the displacement (and traction) vectors that are due to the presence of a small elastic linear crack. The formula reveals that the leading order term is ε ² where ε is the length of the crack, and the ε ³-term vanishes. We obtain an asymptotic expansion of the elastic potential energy as an immediate consequence of the boundary perturbation formula. The derivation is based on layer potential techniques. It is expected that the formula would lead to very effective direct approaches for locating a collection of small elastic cracks and estimating their sizes and orientations.     

Modulation theory for the steady forced KdV–Burgers equation and the construction of periodic solutions

We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg–de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger’s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).     

Concentration of Solutions for Some Singularly Perturbed Mixed Problems: Existence Results

In this paper,we study the asymptotic behavior of some solutions to a singularly perturbed problem with mixed Dirichlet and Neumann boundary conditions. We prove that, under suitable geometric conditions on the boundary of the domain, there exist solutions which approach the intersection of the Neumann and the Dirichlet parts as the singular perturbation parameter tends to zero.     

Stability for Monostable Wave Fronts of Delayed Lattice Differential Equations

This paper is concerned with the stability of traveling wave fronts for delayed monostable lattice differential equations. We first investigate the existence non-existence and uniqueness of traveling wave fronts by using the technique of monotone iteration method and Ikehara theorem. Then we apply the contraction principle to obtain the existence, uniqueness, and positivity of solutions for the Cauchy problem. Next, we study the stability of a traveling wave front by using comparison theorems for the Cauchy problem and initial-boundary value problem of the lattice differential equations, respectively. We show that any solution of the Cauchy problem converges exponentially to a traveling wave front provided that the initial function is a perturbation of the traveling wave front, whose asymptotic behaviour at \(-\infty \) satisfying some restrictions. Our results can apply to many lattice differential equations, for examples, the delayed cellular neural networks model and discrete diffusive Nicholson’s blowflies equation.     

Bernoulli Variational Problem and Beyond

The question of ‘cutting the tail’ of the solution of an elliptic equation arises naturally in several contexts and leads to a singular perturbation problem under the form of a strong cut-off. We consider both the PDE with a drift and the symmetric case where a variational problem can be stated. It is known that, in both cases, the same critical scale arises for the size of the singular perturbation. More interesting is that in both cases another critical parameter (of order one) arises that decides when the limiting behaviour is non-degenerate. We study both theoretically and numerically the values of this critical parameter and, in the symmetric case, ask if the variational solution leads to the same value as for the maximal solution of the PDE. Finally we propose a weak formulation of the limiting Bernoulli problem which incorporates both Dirichlet and Neumann boundary condition.     

Determining the boundary layers in the plane problem for three-layer strips. Part 1

We propose an exact solution of the problem on a boundary layer (a stress-strain state decreasing away from the boundary) for three-layer strips (rods) whose layers are made of different materials. We use the asymptotic integration method to obtain boundary eigenfunctions and a characteristic equation for the parameter describing the boundary layer decay rate. We study how the middle layer material affects the boundary layer extent.     

Computer extended series for a thermally driven gas centrifuge

Linearized, multidimensional, thermally driven flow in a gas centrifuge can be approximately described in regions away from the ends by Onsager's homogeneous pancake equation.¹ Upon reformulation of the general problem, we find a new, simple and rigorous closed form, analytical solution by assuming a special separable solution and replacing the usual Ekman end cap boundary conditions with idealized impermeable, free slip boundary conditions. Then the flow may be described by an ordinary differential equation with solutions in terms of simple, classical functions. By identifying a small parameter, say ?, defining the semi-long bowl approximation, and assuming a power series expansion in ?, a sequence of asymptotic approximations to the master potential is obtained. Not surprisingly, the leading order term involves the well known ‘long bowl’ solution. Using the so-called ‘solving’ property of the 1-D pancake Green's function,² we determine the next higher order solution. This recursive process is carried out on the computer to find all the terms up to O(?⁴). Consequently, the solution of some complex rotating, viscous, heat conducting flow problems that normally require large mainframe computers can be better understood.     

Asymptotic theory of initial value problems for nonlinear perturbed Klein-Gordon equations

Introduction InthispaperasymptotictheoryofthefollowinginitialvalueproblemforanonlinearKlein Gordonequationisconsidered.tt-Δ =εF(t,x,,ε),t>0,x∈R2,(0,x,ε)=0(x,ε),t(0,x,ε)=1(x,ε),x∈R2,(1)where(t,x)isarealvaluedunknownfunction,Δ=2i     

A difference method for singular perturbation problem of hyperbolic-parabolic partial differential equation

In this paper we constructed an exponentially fitted difference scheme for singular perturbation problem of hyperbolic-parabolic partial differential equation. Not only do we take a fitting factor in the equation, but also we put one in the approximation of second initial condition. By means of the asymptotic solution of singular perturbation problem we proved the uniform convergence of this scheme with respect to the small parameter.